Lower bounds for the dyadic Hilbert transform
Language
en
Article de revue
This item was published in
Annales de la Faculté des Sciences de Toulouse. Mathématiques. 2018, vol. 27, p. 265-284
Université Paul Sabatier _ Cellule Mathdoc
English Abstract
In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form $\norm{\Sha f}_{L^2(K)}\geq C(I,K)\norm{f}_{L^2(I)}$where $I$ and $K$ are two dyadic intervals and $f$ supported in $I$. If ...Read more >
In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form $\norm{\Sha f}_{L^2(K)}\geq C(I,K)\norm{f}_{L^2(I)}$where $I$ and $K$ are two dyadic intervals and $f$ supported in $I$. If $I\subset K$ such bound exist while in the other cases $K\subsetneq I$ and $K\cap I=\emptyset$ such bounds are only available under additional constraints on the derivative of $f$. In the later case, we establish a bound of the form $\norm{\Sha f}_{L^2(K)}\geq C(I,K)|\scal{f}_I|$ where $\scal{f}_I$is the mean of $f$ over $I$. This sheds new light on the similar problem for the usual Hilbert transform that we exploit.Read less <
English Keywords
Haar Shift
Dyadic Hilbert transform
ANR Project
Analyse Variationnelle en Tomographies photoacoustique, thermoacoustique et ultrasonore - ANR-12-BS01-0001
Initiative d'excellence de l'Université de Bordeaux - ANR-10-IDEX-0003
Initiative d'excellence de l'Université de Bordeaux - ANR-10-IDEX-0003
Origin
Hal imported